Applications of generalized universal valuations.

We introduce a generalization of the universal valuation semiring defined by Jeffrey and Noah Giansiracusa. We then explicitly characterize the additive structure of this semiring and show that, when applied to Q , this characterization gives the non-Archimedean case of Ostrowski's theorem. We conclude with examples of non-commutative valuations and their applications, such as the detection of the existence of representations of rings in ultrametric vector spaces. [ABSTRACT FROM AUTHOR]